A Derivative Test of Cosmic Growth Saturation

We introduce a derivative-based diagnostic to probe the redshift evolution of cosmic structure growth. While standard ΛCDM cosmology predicts a smooth and monotonic decline of the observable fσ₈(z), we test for deviations in the form of a flattening or reduced slope in its derivative: D(z) = d(fσ₈)/dz We define a quantitative flattening condition over a redshift interval Δz: |D(z)| < ε where ε is determined by observational uncertainties. Such a flattening would directly indicate a departure from purely monotonic growth models. We show that this behavior arises naturally in the presence of a retentive structural contribution, leading to a saturation regime in which growth is no longer governed solely by expansion dynamics. The proposed test provides a minimal and falsifiable distinction between smooth dissipative models and structurally constrained alternatives. The diagnostic can be directly applied to current and upcoming datasets, including DES, Euclid, and LSST.